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Induced current in a wire

Sal determines the current and EMF induced in a wire pulled through a magnetic field. Created by Sal Khan.

Video transcript

Let's say I have a magnetic field popping out of this video. So these little brown circles show us the tips of the vectors popping out of our screen. And in that magnetic field I have this wire, this off-white colored wire. And sitting on that off-white colored wire I have a charge of charge Q. Let me write down the other stuff. So this is the magnetic field B coming out. Let's say I were to take this whole wire. And let's say that the wire overlaps with the magnetic field a distance of L. So let's say the magnetic field stops here and stops here. And let's say that this distance right here, that distance is L. I drew it a little bit weird but you get the idea. From here to here is L. I have this charge sitting on some type of conductor that we can consider a wire. And the magnetic field is pushing out of the page. So in this current formation, let's say I don't have any voltage across this wire or anything. What's going to happen? Well, if I just have a stationary charge sitting in a magnetic field, nothing really is going to happen, right? Because we know that the force due to a magnetic field is equal to the charge times the cross product of the velocity of the charge and the magnetic field. If this wires are stationary and there's no voltage across it, et cetera, et cetera, the velocity of this charge is going to be 0. So if the velocity is 0, we know that the magnitude of a cross product is the same thing as--. So Q is, that's just a scalar quantity-- so that's just Q-- times the magnitude of the velocity times the magnitude of B times sine theta. And in these situations where anything that's going on in this plane is going to be perpendicular to this magnetic field--. So the angle between the magnetic field and any velocity-- if there were any within this plane-- would be 90 degrees. So you wouldn't have to worry about the sine theta too much. But we see if the velocity is 0 or the speed is 0, the magnitude of the velocity is 0, that there's not going to be any net force due to the magnetic field on this charge. And nothing interesting's going to happen. But let's do a little experiment. What happens if I were to move this wire, if I were to shift it to the left, with the velocity V? So I take this wire and I shift it to the left with the velocity V. All right, so the whole wire is shifting to the left. Well, if the whole wire is shifting to the left, this charge is sitting on that wire, so that charge is also going to move to the left with the velocity V. And now things get interesting. The charge is moving to the left with the velocity V, so now we can apply the first magnetism formula that we learned. We could apply this formula. So what's going to happen to this charge? Well, the force of the charge is going to be the charge times the magnitude of the velocity cross the magnetic field vector. So we know that there's going to be some net force. This is non-zero now. And this is non-zero, we're assuming. And we're assuming the charge is non-zero. So what direction is the force going to be in? So let's do our right hand rule on the cross product. V cross B will give us the direction. So point your index finger in the direction of the velocity. I have to look at my own hand to make sure I'm doing it right. So you point your index finger in the direction of the velocity. Point your middle finger in the direction of the magnetic field. The magnetic field is popping out of the page, so your middle finger is actually going to be popping out of the page. Your next two fingers are just going to do something like that. So you're kind of approximating like you're shooting a gun. And then what's your thumb going to do? Your thumb is going to point straight up. This is the palm of-- that's your thumb. This could be your nail, fingernail, fingernail of your thumb, fingernail of your middle finger. This is the direction of the velocity. Let me get a suitable color. The velocity is that way. The magnetic field is popping out of the page. So the force on the particle-- on this charged particle or on this charge-- due to the magnetic field is going to go in the direction of your thumb. So the direction of the force is in this direction. So what's going to happen? There's going to be a net force in this direction on the charge. And the charge is going to move upwards, right? I mean, when you start having a moving-- you could imagine also that you had multiple charges, right? If you had multiple charges here and you're moving the whole wire, all of those charges are going to be moving upwards. And what is another way to call a bunch of moving charges along a conductor? Well, it's a current, depending on how much charge is moving per second. So at least in very qualitative terms, you see that when you move a wire through a magnetic field or when you move a magnetic field past a wire, right? Because they're kind of the same thing, it's all about the relative motion. But if you move a wire through a magnetic field, it is actually going to induce a current in the wire. It's going to induce the current in the wire, and actually this is how electric generators are generated. And I'll do a whole series of videos on how you-- you know, if you're using coal or steam or hydropower, how that turns, essentially that turns these generators around and it induces current. And that's how we get electricity from all of these various energy sources that essentially just make turbines turn. But anyway, let's go back to what we were doing. So let me ask you a question. If this particle-- and this all has a point-- if this particle starts at the beginning. Let's say the particle is right here. So it starts right where the magnetic field starts affecting the wire. And how much work is going to be done on the particle by the magnetic field? Well, what's work? Work is equal to force times distance, where the force has to be in the same direction as the distance, right? Force times distance, I won't mess with the vectors right now. But they have to be in the same direction. So how much work is going to be done on this particle? So the work is going to be the net force exerted on the particle times the distance. Well, this distance is L, right? We say, once a particle gets here there's no magnetic field up here, so the magnetic field will stop acting on it. So the total work done: Work, which is equal to force times distance, is equal to-- so the net force is this up here. Q-- and I'll leave some space-- V cross B times the distance. And the distance right here is just a scalar quantity, so we could put it out front, right? Q times L times V cross B, right? This is-- Q V cross B is the force times the distance. That's just the work done. Now how much work is being done per charge, right? This is how much work is being done on this charge. But let's say there might have been multiple charges, so we just want to know how much work is done per charge. So work per charge. We could divide both sides by charge. So work per charge is equal to this per charge. So it is equal to the distance times the velocity that you're pulling the wire to the left with cross the magnetic field. This is where it gets interesting. So what is work per charge? The units of work are energy, right? Joules. And charge, that's in coulombs. So what are joules per coulomb? This is equal to volts. Volts are joules per coulomb. So this particle, or these charges, are going to start moving in this direction as if there is a voltage difference. As if there is a potential difference between this point and this point. As if this is the positive voltage terminal and this is the minus voltage terminal. So there's actually going to be a voltage-- or a perceived voltage-- difference between this point and this point that will start making the current flow. Let's say you didn't even know that there was a magnetic field here. You would just see this current flowing. You'd be like, oh well, there has to be a voltage difference there, right? But when we're dealing with this-- because when we talk about voltages, that was like a potential difference. That something-- that a particle or a charge has a higher potential energy and that's why it's moving. But it's hard to-- at least for these purposes-- say, well you have a higher potential energy here. It's really being created by the magnetic field. So in this context, people have said that instead of saying that this is creating a voltage difference between this point and this point, if the magnetic field on the moving wire is causing that, people say that it's creating an electromotive force, or an EMF. But EMF, the units are still joules per coulomb or volts. And it really is-- in every way when you're analyzing the circuits-- still the same thing as a potential difference or as a voltage difference. But since it seems a little bit more proactive, it seems like this magnetic field is actually impacting a force on this wire that is causing the current to move. We call it EMF. So we could say that the EMF, the electromotive force-- or the voltage across from here to here, but they're really the same thing-- is equal to the distance of the wire that's in the magnetic field times the velocity-- that you're pulling the wire in-- cross the magnetic field. So let's say, I don't know, let's just throw out a bunch of numbers. Let's say that the magnetic field is-- I'll make it easy-- 2 teslas. My velocity to the left is 3 meters per second. And let's-- just for fun-- let's give this a little bit of a resistance, just so we can figure out something. So let's say this resistance is, I don't know, let's say it is 6 ohms. There's a 6 ohm resistor here. So the resistance of the wire from here to here is 6 ohms. All wires have some resistance. So first of all, what's the EMF? Oh, and let's say that this total distance right here is 12 meters. So the EMF induced on the-- or the electromotive force-- put on to the wire by the magnetic field is going to equal the distance of the wire in the magnetic field-- 12 meters-- times--. Well, when we're just taking the cross product, we know that the velocity is perpendicular to the magnetic field. So we don't have to worry about sine theta because theta is already 90 degrees. So we just have to worry about the magnitudes. So it's going to be 12 meters times the velocity, which is 3 meters per second, times the magnetic field, or the magnitude of the magnetic field, that's 2 teslas. And so the EMF is 12 times 3 times 2. 12 times 6. Which is 72. You could say 72 volts, or 72 joules per coulomb. And now you have that potential difference, or that EMF, across a 6 ohm resistor, right? So that, you just go back to voltage is equal to IR. Or you could write EMF is equal to IR. So EMF divided by resistance. So if we take this EMF and we divide it by the resistance-- divided by 6 ohms-- we get the current, right? EMF divided by resistance is equal to current. So you divide 72 volts or 72 joules per coulomb divided by 6 ohms. And then you get a current going along this wire right here, due to the EMF, due to the magnetic field-- I know it's very messy at this point-- of 12 amperes. Anyway, I'm all out of time. I'll see you in the next video.